Abstract
The depth statistic was defined by Petersen and Tenner for an element of an arbitrary Coxeter group in terms of factorizations of the element into a product of reflections. It can also be defined as the minimal cost, given certain prescribed edge weights, for a path in the Bruhat graph from the identity to an element. We present algorithms for calculating the depth of an element of a classical Coxeter group that yield simple formulas for this statistic. We use our algorithms to characterize elements having depth equal to length. These are the short-braid-avoiding elements. We also give a characterization of the elements for which the reflection length coincides with both depth and length. These are the boolean elements.
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AW is partially supported by NSA Young Investigators Grant H98230-13-1-0242.
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Bagno, E., Biagioli, R., Novick, M. et al. Depth in classical Coxeter groups. J Algebr Comb 44, 645–676 (2016). https://doi.org/10.1007/s10801-016-0683-9
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DOI: https://doi.org/10.1007/s10801-016-0683-9